Question: Simplify the following expression and state the condition under which the simplification is valid: $p = \dfrac{q^2 - 9q + 8}{q^2 - 8q}$
First factor the expressions in the numerator and denominator. $ \dfrac{q^2 - 9q + 8}{q^2 - 8q} = \dfrac{(q - 1)(q - 8)}{(q)(q - 8)} $ Notice that the term $(q - 8)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(q - 8)$ gives: $p = \dfrac{q - 1}{q}$ Since we divided by $(q - 8)$, $q \neq 8$. $p = \dfrac{q - 1}{q}; \space q \neq 8$